Nonparametric maximum likelihood estimators (MLEs) in inverse problems often have non-normal limit distributions, like Chernoff's distribution. However, if one considers smooth functionals of the model, with corresponding functionals of the MLE, one gets normal limit distributions and faster rates of convergence. We demonstrate this for a model for the incubation time of a disease. The usual approach in the latter models is to use parametric distributions, like Weibull and gamma distributions, which leads to inconsistent estimators. Smoothed bootstrap methods are discussed for constructing confidence intervals. The classical bootstrap, based on the nonparametric MLE itself, has been proved to be inconsistent in this situation.

翻译：逆问题中的非参数最大似然估计（MLE）常具有非正态极限分布，例如Chernoff分布。然而，若考虑模型的光滑泛函及其对应的MLE泛函，则可获得正态极限分布和更快的收敛速度。我们以疾病潜伏期模型为例进行论证。在此类模型中，通常采用参数分布（如Weibull分布和Gamma分布）的方法会导致估计量不一致。本文探讨了用于构建置信区间的平滑自助法。经典自助法基于非参数MLE本身，已被证明在此情形下不一致。